Abstract
We prove that the non-increasing rearrangement of a dyadic \(A_1\) weight \(w\) with dyadic \(A_1\) constant \(\big [w\big ]^{\mathcal {T}}_1=c\) with respect to a tree \({\mathcal {T}}\) of homogeneity \(k\), on a non-atomic probability space, is a usual \(A_1\) weight on \((0,1]\) with \(A_1\)-constant \([w^*]_1\) not more than \(kc-k+1\). We prove also that the result is sharp, when one considers all such weights \(w\).
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Acknowledgments
The author would like to thank professor A. Melas for suggesting the problem in this paper. This research has been co-financed by the European Union and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF), Aristeia Code: MAXBELLMAN 2760, Research code: 70/3/11913.
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Nikolidakis, E.N. Dyadic \(A_{1}\) Weights and Equimeasurable Rearrangements of Functions. J Geom Anal 26, 782–790 (2016). https://doi.org/10.1007/s12220-015-9571-0
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DOI: https://doi.org/10.1007/s12220-015-9571-0